Internal Temperature of Skin when Surface Temperature Is Controlled with an Electromagnetic Beam

We study the thermal effect on skin exposed to an electromagnetic beam of time-dependent power. We consider two types of beam power time schedules. In the controlled temperature exposure, the skin surface temperature is increased quickly to a prescribed level using a high beam power; then the surface temperature is maintained at the prescribed level by adjusting the beam power adaptively. In the constant power exposure, the applied beam power is relatively low and stays unchanged over the time. We start both types of exposures at the same time and compare their internal temperatures of skin when they have the same surface temperature. In a non-dimensionalized formulation, we show that at the moment when both exposure types reach the same prescribed surface temperature level, the controlled temperature exposure has a higher internal temperature at all depths. This conclusion is mathematically rigorous and is independent of skin material properties.


Introduction
In many medical applications, such as cancer hyperthermia, patients are exposed to radiofrequency (RF) radiation [1]. We consider the thermal effect of RF radiation on human skin. The electromagnetic energy deposited by RF radiation increases the skin temperature. The skin surface temperature can be measured in In this mathematical study, we allow the beam power to vary with time. We consider two types of beam power time schedules. In the controlled temperature exposure, the skin surface temperature (at the beam center) is increased quickly to a prescribed level using a high beam power; then the surface temperature is maintained at the prescribed level by adjusting the beam power adaptively, leading to a time-varying beam power. The second type of exposure is the constant power exposure, in which the applied beam power is relatively low and stays unchanged over the time. We study the surface and internal temperatures of skin caused by these two types of exposures. We start both types of exposures at the same time. Due to its relatively higher initial power level, the controlled temperature exposure increases the temperature faster in the initial phase. Upon reaching the prescribed surface temperature level, the power is adaptively lowered to maintain the surface temperature. The constant power exposure, on the other hand, increases the temperature relatively slower but steadily. Eventually exposure types reach the prescribed surface temperature level. The main objective of this study is to compare the internal temperatures of the two exposures.
The rest of the article is organized as follows. In Section 2, we discuss the mathematical formulation and solution for the case where the beam power varies with time. Based on the solution for time-varying beam power, in Section 3 we develop the mathematical scheme for adjusting the beam power to maintain the surface temperature at the prescribed level. We run simulations to implement the control scheme and to demonstrate numerically that the beam power is a decreasing function of time in the controlled temperature exposure. This observation motivates Theorem 1. The internal temperatures of the two exposures are examined numerically in Section 4. A key observation is that when both exposures have the same surface temperature, the controlled temperature exposure always has the higher internal temperature at all depths. This finding motivates Theorem 2. In Section 5, we prove Theorem 1 and Theorem 2 rigorously in a dimensionless formulation. Thus, the main conclusions in this study are independent of skin material properties and independent of prescribed temperature level.

Mathematical Formulation for an Electromagnetic Beam of Time-Dependent Power
, T z t denote the skin temperature along the beam center line as a function of depth z and time t. We assume 1) the electromagnetic beam is perpendicular to the skin surface (i.e., beam incident angle = 0); 2) before the exposure, the skin has a uniform initial temperature base T (the baseline temperature); and 3) heat conduction is included only in the depth direction [8] (which is justified given the small length scale of electromagnetic wave penetrating in the skin depth direction and the much larger length scale of beam cross-section, and which allows us to separate z from ( ) , x y ).
The temperature distribution ( ) , T z t is governed by the heat equation where • m ρ is the mass density of the skin; • p C is the specific heat capacity of the skin; • k is the heat conductivity of the skin; • µ is the absorption coefficient of the skin for the beam frequency; • ( ) P t is the beam center power density absorbed into the skin at time t.
We first non-dimensionalize variables and functions in (1). The depth scale is provided by, 1 µ , which describes the characteristic scale of electromagnetic energy penetrating in the depth direction. The time scale is derived from the length scale and parameters of heat capacity and heat conduction. The temperature scale is usually set based on the objective of tests. For example, in studying heat-induced withdrawal reflex, the temperature scale is set to the difference between the activation temperature of nociceptors ( act T ) and the baseline temperature of skin ( base T In this study, we analyze the non-dimensional system (2) and its solution.
Next, we use the temperature solution (3) to design the beam power schedule ( ) P t for controlling the skin surface temperature.

Surface Temperature Control
In this section, we study the controlled temperature exposure. Let T ∆ be the prescribed surface temperature level. The ∆ notation stems from that it is the intended non-dimensional surface temperature rise over the non-dimensional baseline temperature (0). Based on the temperature solution given in (3), we write the surface temperature as The asymptotic behavior of ( ) b t follows from the asymptotic expansion of ( ) When the beam power is kept at any fixed value, ( ) C P t P ≡ , the surface temperature is proportional to the beam power and increases monotonically with time without bound.
In the controlled temperature exposure, we start with a relatively high beam power 0 P . We keep the beam power at 0 P until time 0 t when the surface temperature reaches the prescribed T ∆ . Mathematically, 0 t is governed by For 0 t t > , the beam power is adjusted adaptively to maintain the surface temperature at the prescribed T ∆ . To mimic the realistic experimental situation, we consider the case of adjusting the beam power in discrete time steps. We use a uniform grid for In the controlled temperature exposure, the initial beam power 0 P and the target temperature level T ∆ are prescribed as the specified parameters. In comparison, the initial exposure period 0 t and the subsequent beam power levels { } First, we solve for 1 P in equation In general, when all preceding power levels { } 0 1 , , , j P P P  are known, we solve for It is worthwhile to compare the mathematical control described above and the We first explore numerically the behavior of the beam power time schedule In other words, in the controlled temperature exposure, the beam power schedule ( ) P t is a decreasing function of time.
This theorem is a key analytical tool when we compare the skin internal temperature for the two types of exposures: controlled temperature exposure vs constant beam power exposure. Although Theorem 1 is confirmed numerically in Figure 1 for 2 T ∆ = and 0 8 P = , we will prove it rigorously in Section 5 for all values of T ∆ and 0 P . In Section 4, we compare the skin internal temperature of the two exposure types and summarize the key result in Theorem 2, which is also proved rigorously in Section 5.

Skin Internal Temperature of Controlled Temperature Exposure vs Constant Power Exposure
In the constant power exposure [9], a relatively low beam power L P is applied where j P is solved sequentially from Accordingly, the corresponding skin internal temperature of these two exposure We are interested in comparing the skin internal temperature for the two exposure types at time L t .
Again, we first explore it numerically. We use 0 8 P = and 2 T ∆ = for the controlled temperature exposure (the same parameters used in Figure 1  Theorem 2 When the surface temperature of the constant (low) power exposure reaches the prescribed ΔT, both exposure types have the same surface temperature and the controlled temperature exposure always has a higher internal temperature.
We will prove Theorems 1 and 2 rigorously in the next section.

Proof of Theorems 1 and 2
An analytical expression of ( ) , T z t is given in (3). We rewrite it as   1 e e d Substituting these two terms back into (3) Here we have used ( ) Using (17) at z and at z = 0, we calculate the time derivative of ( ) ( ) ( ) Here we have used Property 1 to conclude that the term over the underbrace is positive.
Graphs of ( ) b t and ( ) tb t are illustrated in the left panel of Figure 3. The right panel of Figure 3 compares ( ) , g z t vs z for several values of t (Property 2). With the results itemized above, we examine the derivative of ( ) Here we have used items 3 and 2 above.  Proof. We examine the derivative of ( )

Proof of Theorem 1
We need to show 0 1 2 P P P < < < . Since our main focus is on the relation among { } j P , we like to write 1 j P + in terms of the preceding power levels. For 1 P , we take the difference between (10) and (8)  It follows that ( ) is defined in (18). For 2 P , we take the difference between (11) and (10).
Here we have used Property 6 and Equation (19) to conclude ( ) ( ) For P 3 , we take the difference between (12) and (11), P P P P t t P P P P t t Here we have used ( )

Proof of Theorem 2
We need to show In conclusion, when both exposure types reach the same surface temperature, the controlled temperature exposure always has a higher internal temperature.

Concluding Remarks
In this study, we considered the thermal effect on skin exposed to an electromagnetic beam. We investigated the skin surface temperature and internal temperature caused by the beam. Specifically, two exposure types were examined. In the controlled temperature exposure, a high beam power is used to increase the skin surface temperature quickly to a prescribed level. Then the beam power is adjusted adaptively to maintain the surface temperature at the prescribed level.
In the constant power exposure, a relatively low beam power is applied without any change in power level over the time. We start both types of exposures at the same time. The controlled temperature exposure will reach the prescribed surface temperature level first since it has a higher initial beam power. To maintain the surface temperature once the prescribed level is attained, beam power drops significantly and keeps declining gradually over the time. When both types of exposures reach the same surface temperature, the controlled temperature exposure always has a higher internal temperature at all depths of skin. We proved this conclusion rigorously in a dimensionless formulation. This conclusion is independent of skin material properties, initial beam power levels and the prescribed surface temperature level.